A retrial model in a nonstationary regime (Q1814814)

One considers a single channel system with arrivals generated by a Poisson process. These arrivals are called primary customers. If the service area is free when a customer arrives, the service starts immediately. If the customer finds the channel busy, he will reapply for service after a random period. Customers standing by are said to be in orbit. When a customer arrives into the orbit he must join a queue with FCFS discipline. The time between repetitions, when the orbit is nonempty, is exponentially distributed and the flows of primary customers and repeated attempts are mutually independent. The service time distribution is assumed to be the same for both types of customers. The paper studies the behaviour of the system in nonstationary case by using the theory of semiregenerative processes. It is assumed that the initial moment is departure instant and there is an arbitrary number of initial customers in orbit. Let \(C(t)\) represent the number of busy channels and \(N(t)\) the number of customers in the orbit at time \(t\). The paper gives a formula for the joint generating function of the process \((C(t),N(t))\) and integral estimation for the difference between blocking probabilities (i.e. the server is occupied) in stationary and nonstationary regimes.